JOURNAL BROWSE
Search
Advanced SearchSearch Tips
ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS
Yun, Jae Heon;
  PDF(new window)
 Abstract
In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.
 Keywords
relaxation iterative method;semi-convergence;pseudo-spectral radius;singular saddle point problem;scaled preconditioner;
 Language
English
 Cited by
 References
1.
M. Arioli, I. S. Duff, and P. P. M. de Rijk, On the augmented system approach to sparse least squares problems, Numer. Math. 55 (1989), no. 6, 667-684. crossref(new window)

2.
Z.-Z. Bai, G. H. Golub, and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004), no. 1, 1-32. crossref(new window)

3.
Z.-Z. Bai, B. N. Parlett, and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005), no. 1, 1-38. crossref(new window)

4.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.

5.
Z. Chao and G. Chen, Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems, Appl. Math. Lett. 35 (2014), 52-57. crossref(new window)

6.
Z. Chao, N.-M. Zhang, and Y.-Z. Lu, Optimal parameters of the generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 266 (2014), 52-60. crossref(new window)

7.
H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput. 20 (1999), no. 4, 1299-1316. crossref(new window)

8.
H. C. Elman and D. J. Silvester, Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 17 (1996), no. 1, 33-46. crossref(new window)

9.
B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen, Minimum residual methods for augmented systems, BIT 38 (1998), 527-543. crossref(new window)

10.
G. H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), no. 3, 71-85. crossref(new window)

11.
F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incom-pressible flow of fluid with free surface, Phys. Fluids 8 (1965), 2182-2189. crossref(new window)

12.
J.-I. Li and T.-Z. Huang, The semi-convergence of generalized SSOR method for singular augmented systems, High Performance Computing and Applications, Lecture Notes in Computer Science 5938 (2010), 230-235.

13.
G. H. Santos, B. P. B. Silva, and J.-Y. Yuan, Block SOR methods for rank deficient least squares problems, J. Comput. Appl. Math. 100 (1998), no. 1, 1-9. crossref(new window)

14.
S. Wright, Stability of augmented system factorization in interior point methods, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 191-222. crossref(new window)

15.
S.-L. Wu, T.-Z. Huang, and X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (2009), no. 1, 424-433. crossref(new window)

16.
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.

17.
J.-Y. Yuan and A. N. Iusem, Preconditioned conjugate gradient methods for generalized least squares problem, J. Comput. Appl. Math. 71 (1996), no. 2, 287-297. crossref(new window)

18.
J. H. Yun, Variants of the Uzawa method for saddle point problem, Comput. Math. Appl. 65 (2013), no. 7, 1037-1046. crossref(new window)

19.
J. H. Yun, Convergence of relaxation iterative methods for saddle point problem, Appl. Math. Comput. 251 (2015), 65-80. crossref(new window)

20.
G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 219 (2008), no. 1, 51-58. crossref(new window)

21.
G.-F. Zhang and S.-S. Wang, A generalization of parameterized inexact Uzawa method for singular saddle point problems, Appl. Math. Comput. 219 (2013), no. 9, 4225-4231. crossref(new window)

22.
N. Zhang, T.-T. Lu, and Y. Wei, Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math. 255 (2014), 334-345. crossref(new window)

23.
N. Zhang and Y. Wei, On the convergence of general stationary iterative methods for range-Hermitian singular linear systems, Numer. Linear Algebra Appl. 17 (2010), no. 1, 139-154. crossref(new window)

24.
B. Zheng, Z.-Z. Bai, and X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009), no. 5-7, 808-817. crossref(new window)